All Questions
Tagged with legendre-polynomials generating-functions
27
questions
5
votes
2
answers
105
views
Simplify in closed-form $\sum_n P_n(0)^2 r^n P_n(\cos \theta)$
Simplify in a closed form the sum $$S(r,\theta)=\sum_{n=0}^{\infty} P_n(0)^2 r^n P_n(\cos \theta)$$ where $P_n(x)$ is the Legendre polynomial and $0<r<1$. Note that $P_n(0)= 0$ for odd $n$ and $...
- 1,258
0
votes
1
answer
53
views
Morse and Fesbach identity $\sum_{n=0}^{\infty} e^{-(n+\frac{1}{2})|u|} \; P_n(x) =(2\cosh u - 2x)^{-1/2}$
In book called Methods of theoretical physics from Morse and Feshbach, there is identity, which I wanted to prove ($P_n(x)$ are Legendre polynomials):
$$\sum_{n=0}^{\infty} e^{-(n+\frac{1}{2})|u|} \; ...
1
vote
1
answer
62
views
Generating Function for Bivariate Legendre Polynomials?
I am aware of the following standard generating function for single-variable Legendre Polynomials:
$$
\sum\limits_{n=0}^{\infty}P_n(x)z^n = \frac{1}{\sqrt{1-2xz+z^2}}
$$
for $x \in \mathbb{R}, z \in \...
- 55
2
votes
1
answer
80
views
What is the combinatorics meaning of the generating function for Legendre polynomials?
I know the generating function has been a super useful tool when finding the Legendre polynomials (or other special functions), or even used to estimate the static electric potential. In the Physics ...
- 21
1
vote
1
answer
117
views
Prove from the generating function that even index Legendre polynomials are even functions.
At this link: http://www.phy.ohio.edu/~phillips/Mathmethods/Notes/Chapter8.pdf
The author writes that one can prove from the generating function of Legendre polynomials that $P_{2n}$ are all even and $...
- 5,696
2
votes
2
answers
428
views
Bivariate generating function for squared binomial coefficients
I want to find a closed form to the bivariate generating function
$$
G(x, y) = \sum\limits_{i, j} \binom{i+j}{i}^2 x^i y^j.
$$
Ideally, I would prefer a direct approach that is based on the definition ...
- 1,117
4
votes
1
answer
130
views
Series of product of legendre polynomials with shifted degree
I am working on some quantum mechanics and I would love to find a closed expression for the series
$$
S(x,y) = \sum_{l=0}^\infty P_l(x) P_{l+1}(y)
$$
$x,y \in \left[ -1, 1\right]$ and also for its ...
3
votes
1
answer
167
views
Averaged value of product of Legendre Polynomials
Note: the following question comes from Alex Meiburg via Faceboook and was found via his work with the Legendre Polynomials in quantum machine learning.
Let $P_k$ be the $k$-th Legendre Polynomial.
...
- 14.9k
6
votes
1
answer
253
views
$I(x) = -\int_0^1 \frac{1}{z}\ln\left(\frac{1-x z + \sqrt{1-2 x z+ z^2}}{2}\right)\,dz$
Is there a closed form integral for $$I(x) = -\int_0^1 \frac{1}{z}\ln\left(\frac{1-x z + \sqrt{1-2 x z+ z^2}}{2}\right)\,dz$$ for $-1 < x < 1$?
This integral is related to Legendre polynomials ...
- 1,258
2
votes
1
answer
292
views
How do I expand $\frac{1}{\sqrt{ (\boldsymbol{r-r'})^2+a} }$ in legendre polynomials (spherical harmonics)?
Using the generating function for the legendre polynomial: $$ \sum_{n=0}^{\infty} P_{n}(x) t^{n}=\frac{1}{\sqrt{1-2 x t+t^{2}}} $$ It's possible to expand the coulomb potential in a basis of legendre ...
- 419
0
votes
0
answers
133
views
Orthogonality of Legendre Polynomials support
I've recently been introduced to orthogonality of Legendre polynomials, I think I understand the idea behind it; that the integral of the products of two polynomials in the range $(1,-1)$ will equal $...
- 253
1
vote
1
answer
105
views
Re-arranging and integrating a Generating Function of Legendre Polynomials
I have a generating function of Legendre Polynomials given by: $G(x,r)= \sum_{n=0}^\infty P_n(x)r^n = (1-2rx +r^2)^{-1/2}$
My problem is that I'm asked to find $\int_{-1}^1P_n(x)dx$ but all I have is $...
- 253
3
votes
1
answer
2k
views
How to prove generating function of legendre polynomials? What am I doing wrong?
I am trying to prove the generating function of Legendre Polynomials:
$$g(x,t) = \frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n$$
By only using the Legendre Differential Equation:
$$
(1-x^2)\...
- 103
1
vote
1
answer
123
views
Legendre polynomial expansion
The known magnetic field of a simple loop $(B_z,B_r)$ is expressed as a combination of constant current I and the 2 Elliptic integrals K(k), E(k), k being related to the cylindrical position $(z,r)$ ...
- 11
5
votes
2
answers
461
views
The coefficient and asymptotic in generating function
Let $L_n$ be the set of all the paths from $(0,0)$ to $(n,0)$ such that every step is $u=(1,1)$ , $d=(1,-1)$ and $r=(2,0)$.
Notice that the path could go under the $x$ axis.
a. Write a generating ...
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